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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Two-sided evaluations based on the variational formulations of integral equations for the stability of elastic rods

Vestnik MGSU 10/2014
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Мoscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 41-47

The author considers the method of two-sided evaluations in solving the problems of stability of one-span elastic non-uniformly compressed rod with variable longitudinal bending rigidity in case of different classic conditions of fixation of the rod ends. The minimum critical value of the loading parameter for the rod is represented as a problem of calculating minimum value of the functional corresponding to the Euler equation, which is the same as the integral equation for the rod stability. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, the author constructs two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones for the required value of the loading parameter. The basic functions here are the derivative forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces applied at the rod ends. The calculation of the lower bounds value is reduced to the determination of the maximum eigenvalues of block matrices. The elements of the aforesaid matrices are expressed through the integrals of basic functions depending on the type of the fixation of the rod ends. The calculation of the upper bound value is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the modular matrices. It is noted that the obtained upper bound evaluations are not worse than the evaluations obtained by the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2014.10.41-47

References
  1. Kupavtsev V.V. Variatsionnye formulirovki integral'nogo uravneniya ustoychivosti uprugikh sterzhney [Variational Formulations of the Integral Equation of Stability of Elastic Bars]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 9, pp. 137—143. (in Russian)
  2. Rzhanitsyn A.R. Ustoychivost' ravnovesiya uprugikh system [Stability of Equilibrium of Elastic Systems]. Moscow, GITTL Publ., 1955, 475 p. (in Russian)
  3. Alfutov N.A. Osnovy rascheta na ustoychivost' uprugikh system [Fundamentals of the Stability Analysis of the Elastic Systems]. 2-nd edition. Moscow, Mashinostroenie Publ., 1991, 336 p. (in Russian)
  4. Kupavtsev V.V. Bazisnye funktsii metoda dvustoronnikh otsenok v zadachakh ustoychivosti uprugikh neodnorodno-szhatykh sterzhney [Basic Functions for the Method of Two-sided Evaluations in the Problems of Stability of Elastic Non-uniformly Compressed Rods]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 63—70. (in Russian)
  5. Panteleev S.A. Dvustoronnie otsenki v zadachakh ob ustoychivosti szhatykh uprugikh blokov [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestiya RAN. Mekhanika tverdogo tela [News of the Russian Academy of Sciences. Solid Body Mechanics]. 2010, no. 1, pp. 51—63. (in Russian)
  6. Santos H.A., Gao D.Y. Canonical Dual Finite Element Method for Solving Post-Buckling Problems of a Large Deformation Elastic Beam. International Journal Non-linear Mechanics. 2012, vol. 47, no. 2, pp. 240—247. DOI: http://dx.doi.org/10.1016/j.ijnonlinmec.2011.05.012.
  7. Manchenko M.M. Ustoychivost' i kinematicheskie uravneniya dvizheniya dinamicheski szhatogo sterzhnya [Dynamically Loaded Bar: Stability and Kinematic Equations of Motion]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 71—76. (in Russian)
  8. Bogdanovich A.U., Kuznetsov I.L. Prodol'noe szhatie tonkostennogo sterzhnya peremennogo secheniya pri razlichnykh variantakh zakrepleniya tortsov. Soobshchenie 1 [Longitudinal Compression of a Thin-Walled Bar of Variable Cross Section with Different Variants of Ends Fastening (Information 1)]. Izvestiya vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction]. 2005, no. 10, pp. 19—25. (in Russian)
  9. Bogdanovich A.U., Kuznetsov I.L. Prodol'noe szhatie tonkostennogo sterzhnya peremennogo secheniya pri razlichnykh variantakh zakrepleniya tortsov. Soobshchenie 2 [Longitudinal Compression of a Thin-Walled Core of Variable Cross Section with Different Variants of Ends Fastening (Information 2)]. Izvestiya vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction]. 2005, no. 11, pp. 10—16. (in Russian)
  10. Selamet S., Garlock M.E. Predicting the Maximum Compressive Beam Axial During Fire Considering Local Buckling. Journal of Constructional Steel Research. 2012, vol. 71, pp. 189—201. DOI: http://dx.doi.org/10.1016/j.jcsr.2011.09.014.
  11. Vo Thuc P., Thai Huu-Tai. Vibration and Buckling Of Composite Beams Using Refined Shear Deformation Theory. International Journal of Mechanical Sciences. 2012, vol. 62, no. 1, pp. 67—76. DOI: http://dx.doi.org/10.1016/j.ijmecsci.2012.06.001.
  12. Kanno Yoshihiro, Ohsaki Makoto. Optimization-bazed Stability Analysis of Structures under Unilateral Constraints. International Journal for Numerical Methods in Engineering. 2009, vol. 77, no. 1, pp. 90—125.
  13. Doraiswamy Srikrishna, Narayanan Krishna R., Srinivasa Arun R. Finding Minimum Energy configurations for constrained beam buckling problems using the Viterbi algorithm. International Journal of Solids and Structures. 2012, vol. 49, no. 2, pp. 289—297.
  14. Rektoris К. Variational methods in Mathematics, Science and Engineering. Prague, SNTL-Publ., Techn. Liter., 1980. (in Russian)
  15. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formuliations of Stability Problems of Elastic Rods Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 285—289. (in Russian)

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Variational formulations of the integral equation of stability of elastic bars

Vestnik MGSU 9/2012
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics 8 (499) 183-46-74, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 137 - 143

The author considers the variational formulations of the problem of stability of non-uniformly
compressed rectilinear elastic bars that demonstrate their variable longitudinal bending rigidity in the
event of different classical conditions of fixation of bar ends.
Identification of the critical bar loading value is presented as a minimax problem with respect
to the loading parameter and to the transversal displacement of the bar axis accompanied by the
loss of stability. The author demonstrates that the critical value of the loading parameter may be formulated
as a solution to the dual minimax problem. Further, the minimax formulation is transformed
into the problem of identification of eigenvalues in the bilinear symmetric and continuous form, which
is equivalent to the identification of eigenvalues of a strictly positive, linear and completely continuous
operator. The operator kernel is presented in the form of symmetrization of the non-symmetric
kernel derived in an explicit form.
Within the framework of the problem considered by the author, the bar ends are fixed as follows:
(1) both ends are rigidly fixed, (2) one end is rigidly fixed, while the other one is pinned, (3) one
end is rigidly fixed, while the other one is attached to the support displaceable in the transverse direction,
(4) one end is rigidly fixed, while the other one is free, (5) one end is pinned, while the other
one is attached to the support displaceable in the transverse direction, (6) both ends are pinned.

DOI: 10.22227/1997-0935.2012.9.137 - 143

References
  1. Rzhanitsyn A.R. Ustoychivost’ ravnovesiya uprugikh system [Stability of the Equilibrium State of Elastic Systems]. Moscow, Gostekhizdat Publ., 1955, 475 p.
  2. Alfutov N.A. Osnovy rascheta na ustoychivost’ uprugikh system [Principles of the Stability Analysis of Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Rektoris K. Variatsionnye metody v matematicheskoy fi zike i tekhnike [Variational Methods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  4. Litvinov V.G. Optimizatsiya v ellipticheskikh granichnykh zadachakh s prilozheniyami k mekhanike [Optimization in Elliptic Boundary-value Problems Applicable to Mechanics]. Moscow, Mir Publ., 1985, 368 p.
  5. Litvinov S.V., Klimenko E.S., Kulinich I.I., Yazyeva S.B. Ustoychivost’ polimernykh sterzhney pri razlichnykh variantakh zakrepleniya [Stability of Polymer Bars in Case of Various Methods of Their Fixation]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 4, vol. 2, pp. 153—157.
  6. Il’yashenko A.V. Lokal’naya ustoychivost’ tavrovykh neideal’nykh sterzhney [Local Stability of Tshaped Imperfect Bars]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 162—166.
  7. Tamarzyan A.G. Dinamicheskaya ustoychivost’ szhatogo zhelezobetonnogo elementa kak vyazkouprugogo sterzhnya [Dynamic Stability of a Compressed Reinforced Concrete Element as a Viscoelastic Bar]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 1, vol. 2, pp. 193—196.
  8. Dudchenko A.V., Kupavtsev V.V. Dvustoronnie otsenki ustoychivosti uprugogo konsol’nogo sterzhnya, szhatogo polusledyashchey siloy [Two-way Estimates of Stability of an Elastic Cantilever Bar, Compressed by a Half-tracking Force]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 1, vol. 6, pp. 302—306.
  9. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formulations of Problems of Stability of Elastic Bars Derived by Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 285—289.
  10. Kupavtsev V.V. O variatsionnykh formulirovkakh zadach ustoychivosti sterzhney s uprugo zashchemlennymi i opertymi kontsami [About the Variational Formulations of Stability Problems for Bars with Elastic Fixation of Supported Bar Ends]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 4, pp. 283—287.

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